Sparta, following the Peloponnesian War, declined remarkably fast - culminating in its defeat at the hands of Thebes at Leuctra in 371BC. By the time Philip II came to power, Sparta's hegemony was over a decade gone. For all intents and purposes, Sparta was irrelevant from Leuctra onwards (bar a brief resurgence during the reign of Cleomenes III (235BC-222BC). While their Laconic response to Philip is famous, it is quite hollow - if Philip or Alexander had sought to subjugate Sparta, it would have been easy - there just wasn't a lot to gain by it when Persia was there for the taking. As you say, Sparta was eventually brought into the League, but not until 331BC, after the disastrous war with Macedon.

As to why the map the Wikipedia article would choose to use is an early one, and why Sparta as a member of the League is overlooked generally, I would suggest that that is because the period of the League's most relevance to greater Hellenic history is the period immediately before and during the invasion of the Achaemenid Empire by Macedon and the League. Following Granicus in 334BC, Alexander even did the following, mocking Sparta for her non-participation in the invasion, "To Athens also he sent 300 suits of Persian armour to be hung up in the Acropolis as a votive offering to Athena, and ordered this inscription to be fixed over them: "Alexander, son of Philip, and all the Greeks except the Lacedaemonians, present this offering from the spoils taken from the foreigners, inhabiting Asia.""

By 331BC, when Sparta was eventually brought into the League, Persia was on her last legs - indeed, that is the year Gaugamela was fought and Darius was murdered. From there on Alexander continued his campaigns in the east, eventually heading into India, and with Persia brought into the Macedonian Empire, the importance and need for the support of the League lessens somewhat. The year after Alexander's death, it was dissolved fully following the Lamian War (322BC). From there the historical focus tends to focus on the big Hellenistic monarchies as well as the burgeoning Roman Republic and Carthage in the west. The Aetolian League and the Achaean League persist for sometime as perhaps successors to the League of Corinth, but with the period of the city-state rapidly drawing to a final conclusion (partially due to depopulation due to settlement in the east), they are only really of great relevance in the context of the Macedonian Wars when they supported larger powers.

I hope this went some way to answering your question.

Sources: The Greeks: History, Culture, and Society, 2nd ed. by Ian Morris and Barry B. Powell, Chapters 17, 19, 20, 22.

Anabasis of Alexander, Arrian, Chapter 16.